In order to design a compensator for a dynamic system, it is necessary to have an effective model for the system. A common approach to modelling a system is in the frequency domain, using Fourier, Laplace or Z transforms. Non-parametric identification is commonly used to obtain an estimate of the frequency response function of a dynamic system. Mathematical models may then be fit to the frequency response function to obtain an analytic model of the dynamic system that generates approximately the same frequency response. This analytic model is often referred to as a transfer function. Strictly speaking, transfer functions refer to the analytic models in the frequency domain of the control system, and frequency response functions refer to a vector of responses of a given control system for a given vector of individual frequencies. The equations in this disclosure are valid for either case. For convenience here, these will all be referred to as transfer functions. Currently, the technology is quite mature for single-input, single-output ("SISO") systems, an example of which is illustrated in FIG. 1. Measurements and curve fits can be made of both analog and digital open-loop and closed-loop systems.
Control design is usually done on an open-loop measurement or model of the physical system of interest. However, many physical systems cannot be measured unless a feedback loop is wrapped around them. This control loop uses some nominal compensator that need only perform well enough to produce a good measurement. From this measurement, an improved model of the physical system is obtained, which allows a better compensator to be designed. This process can be, and often is, iterated until a final compensator is designed.
For single input, single output ("SISO") systems, instruments currently exist that allow one to make a measurement of a closed-loop system and to "unwrap" the closed-loop frequency response function to produce an open-loop frequency response function. The compensator frequency response function, represented by the module C in FIG. 1, is usually easy to compute or measure open-loop, and this function can then be factored out to produce the physical system frequency response function, represented by the module P in FIG. 1. For multi-input, multi-output ("MIMO") systems, closed-loop measurements can be made by procedures analogous to the procedures used for SISO systems. However, no approach has been available for extracting the open-loop frequency response function for the physical system.
In the time-invariant SISO system shown in FIG. 1, a scalar input signal r(t), or its frequency domain representative signal R(f), is received on an input signal line 13 at a first terminal of a difference-forming module 15. The difference module 15 receives a second input signal Y(f) at a second terminal on a second input signal line 17. A difference signal or error signal E(f)=R(f)-Y(f) issues from the difference module 15 on a first intermediate signal line 19, and this signal E(f) is received by a compensator module or signal processor 21. The compensator module 21 issues an intermediate signal U(f) that is received on an intermediate signal line 23 by a physical system or plant module 25 that is being controlled. The plant 25 issues on output signal Y(f) on an output signal line 27 that is also received at the second input terminal of the difference module 15.
Although it is sometimes possible to make measurements of an open-loop system, for example, by use of the signal lines 23 and 27 associated with the physical system or plant module 25 in FIG. 1, it is often difficult to keep the system within its linear operating range without the use of a feedback controller. In a SISO system that has a limited range over which it is linear, it is fairly common to perform measurements of closed-loop systems and then unwrap or open the loop. In FIG. 1, the compensator module 21 and plant module 25 may present either the transfer functions or frequency response functions of these modules. The feedback loop signals may be presented either in continuous-time or in discrete-time. In the continuous-time case, the transfer functions for the modules 21 and 25 are functions of continuous frequency s, and in the discrete-time case, they are functions of discrete frequency z. In either case, the frequency response functions are functions of .omega.=2.pi.f or j.omega.(j=.sqroot.-1). In continuous time s=j.omega.; in discrete time, z=exp(j.omega.T), where T is the sample period. For discrete-time systems, the measurement is usually limited to a frequency span 0.ltoreq.f&lt;(2T).sup.-1, referred to as a Nyquist band.
When making closed-loop measurements are made on a SISO system with an input signal R(f) and output signal Y(f), the ratio between the input signal and the output signal at a given frequency f.sub.0 can be represented by the relation ##EQU1## Incrementing f.sub.0 across the relevant frequency band yields a frequency response function T(f) for the closed-loop system ##EQU2## where the last relationship follows from simple scalar block diagram mathematics. Similar analysis of the error signal E(f)=R(f)-Y(f) produces the relation ##EQU3## If the compensator frequency response function C(O is known, this function can be divided out at each frequency to yield the processor frequency response function P(f).
It is also fairly common to make measurements of MIMO systems of the type shown in FIG. 2. Here the frequency response function matrix describes injection of input signals X.sub.j at the various input terminals and read-out of output signals Y.sub.i at the various output terminals. The frequency response function matrix M, which represents the effect of the module 31, satisfies Y=MX, where the input vector X and output vector Y have the respective dimensions n.sub.1 .times.1 (rows.times.columns) and n.sub.2 .times.1, viz. ##EQU4## The number of reference input signals (n.sub.1 ) and controlled output signals (n.sub.2) for a MIMO system often, but not always, agree (n.sub.1 =n.sub.2) so that the number of input signals or output signals is the same. Note that the compensator/plant/sensor module 31 in FIG. 2 may be an open-loop system itself, in which case an input signal X.sub.j would represent U.sub.j ; or may be a closed-loop system, in which case X.sub.j would represent a reference signal R.sub.j. In the latter case, the module 3 1 can represent one of several general closed-loop maps and the column vector Y of output signals Y.sub.i would be of the same dimension as the column vector X of input signals X.sub.j.
Frequency response analysis and transfer function analysis typically involve an implicit assumption that the system is linear or is operating within its linear region. A few important exceptions exist, such as describing function analysis, but these are not of concern here. If the system nonlinearities are significant, it does make a difference whether the measurement is a series of SISO or single-input, multi-output ("SIMO") measurements or is a single MIMO measurement. However, if the system is operating in its linear region, superposition can be used to synthesize responses to multiple input signals from a series of single input signals. In this linear situation, statistical properties and convenience often determine how the measurement should be carried out.
Some prior work has been performed on measuring SISO and MIMO system parameters and on extracting parametric curve fits from open-loop measurements on such systems. Some work has also been performed on unwrapping closed-loop SISO measurements.
P. L. Lin and Y. C. Wu, in "Identification of Multi-input Multi-output Linear Systems From Frequency Response Data", Trans. ASME, vol. 104 (1982) pp. 58-64, disclose a procedure for determining transfer function parameters for a multiple inputmultiple output linear system, using representation of the transfer function as a numerator frequency polynomial with matrix coefficients divided by a denominator frequency polynomial with matrix coefficients. Output versus input measurements are performed on the system at a sequence of selected frequencies, and the numerator and denominator matrix entries are determined by successive approximations of increasing polynomial degree. It is difficult to determine the magnitude of error present when the polynomials are truncated at selected degrees.
Hewlett Packard Company Application Note 243--2, "Control System Development Using Dynamic Signal Analyzers", 1984, pp. 1-64, discusses the open loop and closed loop models for a SISO linear control system and indicates some of the measurement techniques available at that time (1984) for analysis of a general control system.
E. S.Atkinson et al, in "Low-frequency Analyzer Combines Measurement Capability with Modeling and Analysis Tools", Hewlett Packard Jour., January 1987, pp. 4-16, discusses use of a two-channel Fast Fourier Transform analyzer to measure parameters in a linear SISO control system.
R. C. Blackham et al, in "Measurement Modes and Digital Demodulation for a Low frequency Analyzer", Hewlett Packard Jour., January 1987, pp. 17-25, discusses use of low frequency measurements to determine parameters for a linear SISO control system.
J. L. Adcock, in "Analyzer Synthesizes Frequency Response of Linear Systems" and in "Curve Fitter for Pole-zero System", Hewlett Packard Jour., January 1987, pp. 25-32 and 33-36, discusses use of low pole-zero techniques to determine the numerator and denominator parameters for a linear SISO control system.
R. L. Dailey and M. S. Lukich, in "MIMO Transfer Function Curve Fitting Using Chebyshev's Polynomials", published by TRW Space & Technology Group, 1987 and presented at the 35th Annual Meeting of the Society of Industrial and Applied Mathematics, Denver, Colo., October 1987, disclose an extension of Chebyshev analysis of a SISO system to analysis of an MIMO system. The conventional matrix equation G(s)=N(s)/D(s) for the transfer function is reformulated as an equation G(s)D(s)-N(s)=0 that is linear in all the unknown entries of the matrices D(s) and N(s), where s=j.omega.. This last equation is solved as a set of simultaneous equations, using singular value decomposition techniques applied to Chebyshev polynomial expansions in the frequency .omega. for the matrix functions D(s) and N(s).
Hewlett Packard Company Product Note HP 3562A/3563A-1, "Measuring the Open-loop Frequency Response of the Phase-locked Loop", discusses measurement of PLL parameters for a linear MIMO control system and in other environments.
What is needed is an approach that provides an instrument that extracts MIMO open-loop frequency response functions for a physical system from MIMO closed-loop measurements. Preferably, this approach should provide a non-parametric method of obtaining open-loop responses from closed-loop measurements on a single-input or multiple-input, single-output or multiple-output control system. The method should allow unwrapping of measurements of closed-loop MIMO system to recover the open-loop frequency response function matrix of that MIMO system. The method should be flexible enough to provide measurements of various system parameters, depending upon which system signals are measured, and should extend the capabilities of instruments for unwrapping measurements of SISO systems and those instruments capable of measuring, but not unwrapping, a MIMO loop.